Answers to Quertes. 195 
The principle upon which the solution of such problems as the 
one under consideration is based is that of balancing every gain 
by a corresponding loss. And for the sake of argument let us 
suppose we buy 1,604 articles for £167 1s. 8d., some at PAG eLc: 
The average cost of each is 25 pence. We will reduce all the 
cost prices to pence. If an article worth 1,200 pence is bought 
for 25 pence, there is a gain of 1,175 pence; hence, in order to 
gain one penny, I must buy at of that article (see column 4). 
Now, as I have gained one penny, I must balance this by a loss of 
one penny. So if I pay 25 pence for an article worth 24 pence, 
I lose one penny (column a). Similarly, I find that every time I 
buy one article at 240 pence, I must buy = of an article at 12 
: 1 
pence (column B); or one article at 120 pence, I must buy 5.08 
one at 6 pence (column c); or one article at 60 pence, I must 
buy Pa of one at 3 pence (column pb); or one article at 30 pence, 
I must buy = of one at r penny (column £). Since fractions 
having a common denominator are to each other as their respec- 
tive numerators, I find the common denominators to each of the 
fractions to be 1175, and place the numerators in columns 
F, G, H, I, J, and in adding up these columns they give the 
required number. The only rule for the solution of these 
problems that I know of is given in Brooks’s arithmetic under the 
title ‘‘ Alligation.” 
Philadelphia, Pa. C. Henry Kain. 
409.—Another Rule Wanted.—It is a misconception of the 
scope of “simple arithmetic” that prompts ‘“‘ Puzzled” to ask for 
a “rule” for answering such a question as that which he pro- 
pounds. There are an infinite number of such problems, and 
each would require a different rule. Moreover, to put the rule 
into words, would require about a quarter of a page probably, and 
even then would very likely be difficult to understand. It is, there- 
fore, much better that arithmetical works should confine them- 
selves, as they generally do, to the simple principles of rudimentary 
calculation, leaving problems to be solved by elementary algebra, 
which is easily learnt, and which does the work by appealing 
to the reasoning faculties without burdening the memory with 
countless rules. 
What is called a “simple equation” in algebra readily shows 
the way how to answer “ Puzzled’s” question. This is the arith- 
metical working of it; but how to put it in words as a “ rule” 
would require rather a complicated piece of English composition : 
F rom + deduct a and the product of = and = The remain- 
